The Sun, on the other hand, say Chelsea have not made an approach for Lampard and want Sarri to remain.
Lampard said he would discuss his future with Derby after the club failed to win promotion to the Premier League following their play-off final defeat to Aston Villa last month.
Just as it was with his move from Napoli to Chelsea only a year ago, compensation has proven a sticking point with Sarri still having a year left on his £5.35million-a-season Stamford Bridge contract.
The defender's stock has risen in the market and Maurizio Sarri's impending move to Juventus has triggered speculation that the Italian champions also want to sign Emerson from Chelsea.
There are a lot of opinions about whether or not Lampard would be a good fit as a manager at Stamford Bridge, some say it'll be too much for him to handle in just his second season, and others think he'll be flawless for the job.
It comes after reports Jose Mourinho could make a shock return to Chelsea for a third spell in charge - and after Rio Ferdinand described his former England team-mate Frank Lampard as "the flawless choice" for the Blues job.
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When you get into the particulars: Are the men and women really bringing in equal amounts of revenue? They were world class. Another view was that they should perhaps tone down the celebrations once they were sure they had won the match .
"He is the ideal choice", he told the Mirror .
"But at the end of the day, he finished third behind the two best teams for many, many years in the Premier League, he won the Europa League and was in the final of the League Cup, which they lost on penalties".
"I work and I try not to think about these things".
"I will actually be surprised if they look elsewhere", the BT Sport pundit added. He trained the younger kids at the club. "They have tried with all these great managers".
"Frank knows the club and he knows the players and the players will respect him".
What can Lampard expect if he goes back to Stamford Bridge?
"He has lowered the playing age at Derby - his remit before the season would have been around that". It massively lowers expectations for his first team, meaning he's going to get what amounts to a free hit.